which shows that the length of the vector is unchanged after application of the linear transformation represented by \( V^\T \) so that operation must be a rotation.

Similarly the operation of \( U \) on \( \Sigma V^\T \Bx \) also must be a rotation. The operation \( \Sigma = [\sigma_i]_i \) applies a scaling operation to each component of the vector \( V^\T \Bx \).

All linear (square) transformations can therefore be thought of as a rotate-scale-rotate operation. Often the \( A \) of interest will be symmetric \( A = A^\T \).

where a no-op trace could be inserted in the second order term since that quadratic form is already a scalar. This \( (\spacegrad^2 F)^\T \Delta X \) term has essentially been found implicitly by performing the linear variation of \( \spacegrad F \) in \( \Delta X \), showing that we must have